Differential equations worksheet with answers

Free printable Differential equations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
ty1

A mass–spring–damper is described by y″ + 4.8y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 64y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 8.4y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8.4y′ + 36y = 0
ζ =
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = 5 and y′(0) = -4. Find y(0.5).

y″ + 3y′ + 2y = 0y(0) = 5y′(0) = -4
y(0.5) =
4.Medium
t (min)T (°C)29Ta​27 °C

A cup of coffee at 89 °C is left in a room at 22 °C. After 29 min its temperature is 27 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 89 °CTa = 22 °CT(29) = 27 °C
k = min⁻¹
5.Hard
tyK = 1y(1.5) = ?

A system obeys y″ + 4y′ + 8y = 8 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 4y′ + 8y = 8y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
ty0.5y(0.5) = ?

Solve y′ + y = 6e−0.5t with y(0) = 2. Find y(0.5).

y′ + y = 6e−0.5ty(0) = 2
y(0.5) =

Answer key

  1. 1. ζ = 0.3
  2. 2. ζ = 0.7
  3. 3. y(0.5) = 3.271
  4. 4. k = 0.08949 min⁻¹
  5. 5. y(1.5) = 1.042
  6. 6. y(0.5) = 3.28

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